{
  "abstract": "Discrete gradient methods provide a structure-preserving approach to energy-consistent time integration of dissipative Hamiltonian systems. In their conventional formulation, however, they generally attain at most second-order convergence. Extending the perturbation principle previously developed for Poisson systems to the dissipative setting, we derive fourth-order average vector field discrete gradient schemes. A congruence-based completion preserves the positive semi-definiteness of the dissipation matrix, thereby guaranteeing unconditional energy dissipativity for all positive step sizes. This completion concept provides a systematic route to higher-order energy-consistent schemes. For purely dissipative systems, an alternative exponential completion is derived, together with structure-preserving rational approximations. In particular, we characterize Padé approximants that preserve positive semi-definiteness and can be incorporated into energy-associated splitting schemes. Numerical experiments on a damped physical pendulum and a dissipative Fermi--Pasta--Ulam system demonstrate the importance of the positive semi-definite completion for avoiding nonphysical numerical instabilities and capturing the correct qualitative dynamics. They further show that the proposed methods achieve the desired accuracy and energy consistency at a computational cost competitive with existing Galerkin-type approaches. Moreover, the fourth-order schemes outperform the conventional second-order discrete gradient method in computational efficiency over the considered accuracy range, despite their higher per-step cost.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24692",
  "author_total": 4,
  "authors": [
    {
      "name": "Lucas Lautwein"
    },
    {
      "name": "Nicole Marheineke"
    },
    {
      "name": "Håkon Noren Myhr"
    },
    {
      "name": "Kevin Schäfers"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4vextbb45ye1e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24692",
  "fulltext_url": null,
  "id": "4vextbb45ye1e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24692",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2612|Numerical methods for differential equations"
    ]
  ],
  "openalex": "W7214007193",
  "pdf_url": "https://arxiv.org/pdf/2609.24692",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4vextbb45ye1e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:19Z",
      "native_id": "2609.24692",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24692"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Fourth-Order Energy-Consistent Average Vector Field Discrete Gradient Methods for Dissipative Hamiltonian Systems",
  "title_text": "Fourth-Order Energy-Consistent Average Vector Field Discrete Gradient Methods for Dissipative Hamiltonian Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/4vextbb45ye1e",
  "venue": "arXiv",
  "versions": []
}